Proofs from THE BOOK
Encyclopedia
Proofs from THE BOOK is a book of mathematical proof
s by Martin Aigner
and Günter M. Ziegler
. The book is dedicated to the mathematician
Paul Erdős
, who often referred to "The Book" in which God
keeps the most elegant proof of each mathematical theorem
. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."
Proofs from THE BOOK contains 32 sections (40 in the 4th edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory
, geometry
, analysis
, combinatorics
and graph theory
. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by Karl Heinrich Hofmann. It has gone through four editions in English, and has been translated into Persian
, French
, German
, Hungarian
, Italian
, Japanese
, Chinese
, Polish
, Portuguese
, Korean
, Turkish
, Russian
, and Spanish
.
The proofs include:
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
s by Martin Aigner
Martin Aigner
Martin Aigner is an Austrian mathematician, professor at Freie Universität Berlin since 1974, with interests in combinatorial mathematics and graph theory....
and Günter M. Ziegler
Günter M. Ziegler
Günter M. Ziegler is a German mathematician. Ziegler is known for his research in discrete mathematics and geometry, and particularly on the combinatorics of polytopes.- Biography :...
. The book is dedicated to the mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, who often referred to "The Book" in which God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....
keeps the most elegant proof of each mathematical theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."
Proofs from THE BOOK contains 32 sections (40 in the 4th edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by Karl Heinrich Hofmann. It has gone through four editions in English, and has been translated into Persian
Persian language
Persian is an Iranian language within the Indo-Iranian branch of the Indo-European languages. It is primarily spoken in Iran, Afghanistan, Tajikistan and countries which historically came under Persian influence...
, French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
, German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
, Hungarian
Hungarian language
Hungarian is a Uralic language, part of the Ugric group. With some 14 million speakers, it is one of the most widely spoken non-Indo-European languages in Europe....
, Italian
Italian language
Italian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...
, Japanese
Japanese language
is a language spoken by over 130 million people in Japan and in Japanese emigrant communities. It is a member of the Japonic language family, which has a number of proposed relationships with other languages, none of which has gained wide acceptance among historical linguists .Japanese is an...
, Chinese
Chinese language
The Chinese language is a language or language family consisting of varieties which are mutually intelligible to varying degrees. Originally the indigenous languages spoken by the Han Chinese in China, it forms one of the branches of Sino-Tibetan family of languages...
, Polish
Polish language
Polish is a language of the Lechitic subgroup of West Slavic languages, used throughout Poland and by Polish minorities in other countries...
, Portuguese
Portuguese language
Portuguese is a Romance language that arose in the medieval Kingdom of Galicia, nowadays Galicia and Northern Portugal. The southern part of the Kingdom of Galicia became independent as the County of Portugal in 1095...
, Korean
Korean language
Korean is the official language of the country Korea, in both South and North. It is also one of the two official languages in the Yanbian Korean Autonomous Prefecture in People's Republic of China. There are about 78 million Korean speakers worldwide. In the 15th century, a national writing...
, Turkish
Turkish language
Turkish is a language spoken as a native language by over 83 million people worldwide, making it the most commonly spoken of the Turkic languages. Its speakers are located predominantly in Turkey and Northern Cyprus with smaller groups in Iraq, Greece, Bulgaria, the Republic of Macedonia, Kosovo,...
, Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
, and Spanish
Spanish language
Spanish , also known as Castilian , is a Romance language in the Ibero-Romance group that evolved from several languages and dialects in central-northern Iberia around the 9th century and gradually spread with the expansion of the Kingdom of Castile into central and southern Iberia during the...
.
The proofs include:
- Proof of Bertrand's postulate
- Proof that e is irrationalProof that e is irrationalIn mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...
(also showing the irrationality of certain related numbers) - Five proofs of the infinitude of the primesPrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, including EuclidEuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's and Fürstenberg'sFurstenberg's proof of the infinitude of primesIn number theory, Hillel Fürstenberg's proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences....
.